166 
where S denotes the sum of all the differently arranged pro- 
ducts of m+n factors, of which m— are ys, n—v are 28, w 
are js, and vy ares. Now the number of these arrangements 
in S is 
(m+n)! 
(m—p)! (n—v)l plow? 
and § itself will obviously be of the form NZ (nu, v), N being 
some numerical coefficient depending upon m, n, pw, and v. 
But as the number of differently arranged terms in & (yu, v) is 
(u - v) ! 
piv! ? 
it is plain that we shall have 
(m+n)! 
Nan)! @—) Gey)? 
and consequently, 
min! 
C,= - = 
(m=)! (n—v)! (ut v)! (us ») 
«Thus, we have found that C=(C,, and as this is true for 
the numerical coefficient of every term in the development of 
7 y"2", we are warranted in concluding that this latter ex- 
pression is equal to the 
((m+n)! 
m!\n! 
(m+n)! 
m!n! 
th 
} part of the sum of all the 
differently arranged products of the m+n factors, of which m 
are equal to y+ jr, and z to z+ ke. 
«¢The statement of this theorem, and of other similar ones, 
may be rendered simpler and more elegant by our assigning a 
name and symbol to the last-mentioned expression. I propose 
to call.it the mean value of the product of the factors combined 
in different orders: and for the present to denote it by the 
symbol 
M (y+ju, 2+ ke). 
We may now proceed to interpret the expression 
